| Contents |
| We study the existence of symmetric positive solutions of the generalized H\'{e}non
equation
$$
-\Delta u = f(x)u^p, \quad u>0 \quad \mbox{in } \Omega, \qquad u= 0 \quad \mbox{on } \partial \Omega.
$$
Here $\Omega$ is a bounded domain in $\mathbb{R}^N$, $f\in L^\infty(\Omega)$ and $f(x)$ may be positive or may change its sign. Let $G$ be a closed subgroup of the orthogonal group $O(N)$. We assume that $f(x)$ and $\Omega$ are invariant under the action of $G$, that is, $f(gx)=f(x)$ for all $g\in G$, $x\in \Omega$ and $g(\Omega)=\Omega$ for all $g\in G$. For a proper closed subgroup $H$ of $G$, we prove the existence of a positive solution which is $H$ invariant but $G$ non-invariant under suitable assumptions of $H$ and $G$. |
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